I've always thought that the use of the preposition 'to' with the verb 'convince' is correct. But I recently learned from a reliable source that 'to' shouldn't be used with 'convince' but rather with 'persuade'. So instead of saying for instance 'I should try to convince him to work harder', I'd rather say 'I should try to convince him of his ability to work harder' or 'I should try to convince him that he can work harder' or 'I should persuade him to work harder.'

Grammatically, convince to and convince of are both correct and have different meanings. Why would you have difficulty in using one over the other if the meaning you want to convey is clear? Expressing a thought in different ways, all grammatically correct, is but a matter of style. See related usage in corpora and it becomes clear.
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KrisDec 27 '11 at 4:27

3 Answers
3

Short answer: they're both fine, and you clearly already understand when to use which.

Use of with a noun phrase, to with a predicate, and that for a clause. You gave very good examples of all three in your question. The original reason the to usage was frowned upon is that convince was supposedly only for thoughts and ideas; if something someone said resulted in you actually doing something, then you were persuaded rather than convinced. This rule is rather old. Here's what Merriam-Webster's Dictionary of English Usage has to say:

To sum up: long ago persuade became established in a use connoting mental acceptance without following action -- a sense Richard Grant White thought should be reserved for convince. Persuade still has this use, often with the same of and that constructions regularly found with convince. Sometime around the middle of this century, convince began to be used to connote mental acceptance followed by action, usually in a construction in which an infinitive phrase follows the verb [that is, with to]. This construction is now a fully established idiom. The earlier usage writers who tried to fence off persuade from convince and the later ones who tried to fence off convince from persuade have failed alike. And in another generation perhaps no one will care.

You can read the rest of the entry on page 297 on Google Books. They also give attested usages from the New York Times, etc.

@TeHbTeHb: Sorry, your link is not correct. Need to access through conc.lextutor.ca/concordancers/concord_e.html. Lextutor gives this result for 'convince to': '...he sought to convince another National League club to move here.'
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KrisDec 27 '11 at 4:22

In "convince him to" or "persuade him to", "to" is not a preposition — it's part of the infinitive verb that follows. "Convince him to (verb)" is a legitimate construction. The "of" form takes a noun — "convince him of (some fact/position/etc)".

Examples taken from merrian-webster defition of convince include both the use of of and to:

He convinced me that the story was true.
They convinced us of their innocence.
I managed to convince myself that I was doing the right thing.
We convinced them to go along with our scheme.
I was unable to convince her to stay.

The use of convince to talk about persuading someone to do something
is considered by many British speakers to be wrong or unacceptable. It
would be preferable to use an alternative such as persuade or talk
into.

(may take a clause as object) to make (someone) agree, understand, or realize the truth or validity of something; persuade

Chiefly US to persuade (someone) to do something

Besides this usage issue, there is also, as JohnJamesSmith said, a difference in meaning.
Here is an interesting discussion at WordReference, which gives a very good example, concluding that there are cases where the replacement of convince for persuade just for the sake of correctness, can eventually have an actual degradation in meaning. Let me quote the post:

I actually feel like there's a meaningful difference between convince
someone to do something and persuade someone to do something, at least
in certain contexts.

Since convince, as you say, geostan, involves changing someone's
beliefs or opinions, I feel like that semantic property holds even for
the convince someone to construction. Let me give you an example:

Two friends, Bob and George, get caught stealing from a store. Bob is
always getting into trouble, whereas George is generally a nice boy.
When questioned by his parents, George says, "I know it was wrong to
steal, but Bob convinced me to!"

In this context, convinced implies, at least to me, that George
changed his beliefs, at least temporarily, in order to steal. That is,
for that brief moment, Bob managed to make George believe that it was
okay to steal, or that he could ignore his conscience/beliefs.

Persuaded, on the other hand, would imply that George stole knowing
(and still believing) full well that it was wrong.

It may not be a great example, but if you consider the adjectives
convincing and persuasive, I think the difference is there. We could
say that "Bob was convincing" means that Bob was able to convince
George that stealing was okay, whereas "Bob was persuasive" simply
means Bob was able to get George to steal, without having him change
his beliefs.

So, I would conclude that in most cases it is a matter of BrE or AmE use, but you should always bear in mind that you can not always replace convince with persuade, because both verbs have slightly different meanings.